Day trips from Nice

There are probably dozens of possibilities, but you haven't told us anything about your interests or how you will be traveling (rental car vs. public transportation). It would also be good to know your tolerance for places with really heavy tourist traffic.

We spent two weeks in Nice and had a wonderful time there. It is well positioned for many day trips as buses and trains both travel along the coast. We used buses except for Antibes as train was better for it. Our favorites were Villefranche, Antibes( by train) and Musee Picasso, Cap Ferrat with Villa Ephrussi and Villa Kerylos, St-Paul de Vence and Foundation Maeght.
We also went to Monaco by bus. Buses travel right along the water, while trains are farther back.

Most coastal destinations are served by pretty frequent train and/or bus. Heading inland/uphill may take more connections or planning. Get a few ideas at https://www.ricksteves.com/europe/france/french-riviera.

It really does depend on your interests.... some of these were mentioned above

Villefranche-sur-Mer and Cap Ferrat (by train or bus) and the Villa Ephrussi or the Coastal Walk
Eze (combined with Monaco, perhaps)
St. Paul de Vence (bus 400 from Nice)
Cannes/Antibes combo
Train des Merveilles and Chemin de Fer de Provence/Train des Pignes
Ferry to St. Honorat (Leyrin Islands) or St. Marguerite
Sentier Littoral, Cap d'Antibes
Menton and Ventimiglia, Italy (Friday Market)

If you have a car, you can take a drive in the hills. Common loop includes St Paul, Tourettes-Sur-Loup, Gourdon and the Gorges du Loup. There are also less common destinations like Peillon, Biot, La Turbie, etc

Do get Rick's Provence And The French Riviera book; it has MUCH more detail on this area than his general France book.

My favorites were La Turbie (the view directly down over Monaco is great), Monaco (particularly the gardens, the aquarium, and the Prince's car collection), the gardens at the top of Eze, Juan les Pins, and best of all, Haut des Cagnes (where you can visit Renoir's house).

For trips west of Nice, the train is MUCH faster than the bus, although you have to take a bus to get to Haut des Cagnes. For trips east of Nice, both train and bus can work well; buses can get closer to places like Villa Ephrussi, but trains run later into the night and so you will want these after the buses stop running.

My favourite day trips from Nice:

If you are into twentieth-century art at all, visit the Fondation Maeght art gallery just outside St Paul de Vence, and also try to see the convent chapel at Vence which was decorated by Matisse after one of the sisters nursed him during an illness.

Entrevaux is a very attractive medieval hilltop town to the north of Nice, on the railway line out to Digne. The rail trip up the Var gorge is very scenic as well. Because it's inland, it tends to attract fewer tourists than the comparable places closer to the coast and doesn't get so crowded.